Ordering effect of kinetic energy on dynamic deformation of brittle solids
Introduction
The practical and scientific significance of deformation processes evolving at high strain rates is exceeded only by their complexity; the considered thermodynamic processes are non-stationary, non-local, and far from equilibrium. Despite the truly enormous progress in experimental analysis, there are inherent difficulties, if not limitations, when it comes to extreme loading rates (⩾107 s−1). It is especially difficult to characterize the damage evolution and failure mechanisms due to the difficulty in recovering tested samples. The numerical experiments presented in this study cover the strain-rate range from medium to high, which are commonly explored by the Hopkinson bar and plate-impact experiments. The former is performed (up to 103–104 s−1) under uniaxial stress conditions while the latter (up to 107 s−1) under uniaxial strain conditions. The objective of this investigation is to elucidate the effect of material micro-texture on fracture of dynamically loaded brittle materials. The numerical simulations of the uniaxial tension test, the most common of all mechanical tests for structural materials, appear to be a useful tool for extrapolation of the experimental results.
The idealized brittle material is approximated by a two-dimensional triangular lattice: a Delaunay simplical graph dual to the irregular honeycomb system of Voronoi polyhedra representing, for example, grains of a ceramic material (Krajcinovic, 1996).1 Grain boundaries, the most common examples of weak interfaces in brittle materials, are known to have a profound effect on their structure-sensitive properties such as the dynamic strength. The primary mechanism of damage evolution in polycrystalline ceramic materials is widely reported to be the intergranular microcracking. According to Lawn (1993), grain boundaries “are especially weak in ceramics because of the stringent directionality and charge requirements of covalent–ionic bonds”. Furthermore, as the regions of lower atomic density, grain boundaries act as sources of, and sinks for, structural defects. Finally, sintering and hot pressing at high temperatures followed by cooling causes residual stresses due to the anisotropy of grains (Davidge and Green, 1968, Curtin and Scher, 1990). Therefore, in the present lattice model, the microstructural texture is represented by a network of grain boundaries; the cracking is assumed to be intergranular; and the local stress and strain fluctuations on a scale shorter than the grain facet size are neglected (Krajcinovic, 1996). In more general terms, we may think of lattice models as “a discretization of higher than one-dimensional materials by a network of one-dimensional elements characterized by element constitutive equations, and a breaking condition”, (Jagota and Bennison, 1994). The model incorporates both variability and uncertainty in a straightforward manner. Variability, also termed randomness or aleatory variability, is the natural randomness in the process. Uncertainty, also termed epistemic uncertainty, is the uncertainty in the model; it is due to limited knowledge or limited availability of data or both. The mesoscale material texture is an example of the aleatory variability. The disorder may be topological (unequal coordination number), geometrical (unequal length of bonds), or structural (unequal strength and stiffness of bonds). The disorder is further enhanced by damage evolution, which is governed (to an extent depending on the deformation rate) by the local fluctuations of the energy barriers quenched within the material, and the local fluctuations of stress.
Finally, although the lattice models are used often in modeling the fracture behavior of inhomogeneous or multi-phase systems, it is important to recognize their deficiencies and limits of their applicability. The two-dimensional systems exhibit inherent topological limitations; and the extreme geometrical disorder, which is intentionally used in this study, causes additional side effects. Nonetheless, we believe that the most notable numerical artifacts of lattice models (Jagota and Bennison, 1994, Monette and Anderson, 1994) do not have a first-order effect on universal trends observed in this investigation, keeping in mind that we are dealing with relatively large geometrical and structural disorder and the nucleation-dominated damage evolution modes.
Section snippets
Details of the simulation
The approximation of a material by a particle lattice is inspired by the mesoscale morphology of a certain class of brittle materials, and successes of molecular dynamics method as a tool of solid mechanics (for overview of molecular dynamics, see, for example, Hoover, 1986, Allen and Tildesley, 1994). Within this framework, continuum can be defined as a collection of discrete elements, known as “continuum particles” (Wiener, 1983). Assuming that positions and momenta of particles are known in
Dynamic tensile strength
The stress–strain curves obtained for 30 realizations at strain rates 1 and 1 × 107 s−1 are presented in Fig. 4. Three qualitative observations can be made:
- •
the response is nearly linear up to failure for moderate and the highest strain rates, but not for the transient strain rates characterized by the rapid strength increase;
- •
the loading-rate increase results in increase of the dynamic tensile strength (σm);
- •
the loading-rate increase results in decrease of the strength dispersion.
The maximum dynamic
Summary
The objective of this work is to investigate universal trends in which the disorder and strain rate influence dynamic behavior of the idealized brittle material, by performing repeated lattice simulations for different physical realizations of the microstructural statistics. The dynamic simulations of the uniaxial tension test, , are performed under practically identical stress conditions. The results reveal the ordering effect of the kinetic energy on the dynamic response of
References (26)
- et al.
Experimental and numerical study of concrete at high strain rates in tension
Mech. Mater.
(2001) - et al.
A grain level model for the study of failure initiation and evolution in polycrystalline brittle materials. Part I: Theory and numerical implementation
Mech. Mater.
(2003) Shock-wave compression of brittle solids
Mech. Mater.
(1998)- et al.
Pressure–shear impact and the dynamic viscoplastic response of metals
Mech. Mater.
(1985) - et al.
Statistical models of brittle deformation. Part II: Computer simulations
Int. J. Plasticity
(1999) - et al.
Strain-rate effect on brittle failure in compression
Acta Metall. Mater.
(1994) - et al.
Dynamic compressive strength of silicon carbide under uniaxial compression
Mater. Sci. Eng. A
(2001) - et al.
Computer Simulation of Liquids
(1994) Absence of diffusion in certain random lattices
Phys. Rev.
(1958)- et al.
Brittle fracture in disordered materials: a spring network model
J. Mater. Res.
(1990)
Strength of two-phase ceramic/glass material
J. Mater. Sci.
Micromechanics of Flow in Solids
Local inertial effects in dynamic fragmentation
J. Appl. Phys.
Cited by (12)
Ductile-to-brittle transition of ferritic steels: A historical sketch and some recent trends
2023, Engineering Fracture MechanicsDynamic tensile constitutive equations and fracture of metallic neodymium and lanthanum
2012, Materials Science and Engineering: ACitation Excerpt :Mastilovic et al. [30] studied ordering effect of kinetic energy on dynamic deformation of brittle solids by an ensemble of “continuum particles”. It was indicated [30]: “Increase in loading rate results in transition from random to deterministic damage evolution patterns. This ordering effect of kinetic energy is attributed to diminishing flaw sensitivity of brittle materials with the loading-rate increase”.
Some sigmoid and reverse-sigmoid response patterns emerging from high-power loading of solids
2018, Theoretical and Applied MechanicsTwo-dimensional discrete damage models: Discrete element methods, particle models, and fractal theories
2015, Handbook of Damage Mechanics: Nano to Macro Scale for Materials and StructuresA microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices
2014, ZAMM Zeitschrift fur Angewandte Mathematik und MechanikBottom-up modeling of damage in heterogeneous quasi-brittle solids
2013, Continuum Mechanics and Thermodynamics